16 problems

Fix an integer $c$ prime to $D\cdot N$, let $\mathcal{O}_c$ be the order of the real quadratic field $F$ of conductor $c$, and let $Q\in\mathcal{F}_{Dc^2}$ represent a class of qua…

Let $K_S^{\mathrm{ab}}$ be the maximal abelian extension of $K$ unramified and totally split at the primes in $S$, let $\mathcal G=\operatorname{Gal}(K_S^{\mathrm{ab}}/K)$, and let…

Non-archimedean Darmon point conjecture. The local point $P_\psi$ is rational over $H_{\mathcal O}^+$, and, for every $\alpha\in\operatorname{Pic}^+({\mathcal O})$,

Archimedean Darmon point conjecture. The isogeny $\beta$ can be chosen so that

Oda–Yoshida isogeny conjecture. There exists an isogeny

Darmon point globality conjecture. The isogeny $\beta$ can be chosen so that $P_\psi$ is the image of a global point in $E(H_{\mathcal O}^+)$, and, for every…

Isogeny conjecture. There exists an isogeny

Let $K/F$ be a quadratic extension, let $\mathcal{O}$ be an order in $K$, let $K_{\mathcal{O}}^+$ be the associated narrow ring class field, and let $P_{q,b}$ be the Darmon point a…

Greenberg's rationality conjecture. The local point $P_\psi$ is a global point, and more precisely

Darmon's conjecture. The local point $P_\tau=\Phi_{\operatorname{Tate}}(J_\tau)$ belongs to $E(H_\tau)$, where $H_\tau$ is the ring class field associated with the order determined…

Darmon's conjecture. The point $P_\varphi$ belongs to $E_0(H_{\mathfrak c})$, the point $P_K$ belongs to $E_0(K)$, and $P_K$ is non-torsion if and only if

Yuan–Zhang–Zhang-type conjecture. There exists $b\in\widehat{B}^{\times}$ with $k(b,\omega_{\varphi})\neq0$ such that

Assume that $E(F)$ has rank $1$, choose a generator $P_0$ of $E(F)$ modulo torsion, and for each admissible totally positive $t\in O_F$ let $P_t\in E(F)$ be the trace of the Darmon…

Darmon's reciprocity conjecture. The point $P_b^{\beta}$ lies in $E(K^{\mathrm{ab}})$ and

Let $\mathcal{O}$ be the quadratic order used to define the Darmon points, let $H$ be its narrow ring class field over the real quadratic field $K$, and let $J^{(\ell)}_\pm$ be the…

Let $\ell$ be a prime, let $K_\ell$ be the unramified quadratic extension of $\mathbb{Q}_\ell$, and let $T_\epsilon/L_\epsilon$ be the rigid analytic quotient associated with the s…